The Incompressible Navier-Stokes Equations From Black Hole Membrane Dynamics

TL;DR

This paper demonstrates that the dynamics of black hole horizons in asymptotically Anti-de-Sitter space-time can be described by the incompressible Navier-Stokes equations, linking gravitational horizon behavior to fluid dynamics.

Contribution

It establishes a general connection between black hole membrane dynamics and incompressible Navier-Stokes equations using the Membrane Paradigm approach.

Findings

Horizon dynamics correspond to non-relativistic fluid flow.

The approach applies to general non-singular null hypersurfaces.

Rindler horizon dynamics also follow Navier-Stokes equations.

Abstract

We consider the dynamics of a d+1 space-time dimensional membrane defined by the event horizon of a black brane in (d+2)-dimensional asymptotically Anti-de-Sitter space-time and show that it is described by the d-dimensional incompressible Navier-Stokes equations of non-relativistic fluids. The fluid velocity corresponds to the normal to the horizon while the rate of change in the fluid energy is equal to minus the rate of change in the horizon cross-sectional area. The analysis is performed in the Membrane Paradigm approach to black holes and it holds for a general non-singular null hypersurface, provided a large scale hydrodynamic limit exists. Thus we find, for instance, that the dynamics of the Rindler acceleration horizon is also described by the incompressible Navier-Stokes equations. The result resembles the relation between the Burgers and KPZ equations and we discuss its implications.